DEFUN-MODE

Two ``defun-modes'' are supported, :program and :logic. Roughly
speaking, :program mode allows you to prototype a function for
execution without any proof burdens, while :logic mode allows you to
add a new definitional axiom to the logic. The system comes up in
:logic mode. Execution of functions whose defun-mode is :program
may render ACL2 unsound! See defun-mode-caveat.

When you define a function in the ACL2 logic, that function can be
run on concrete data. But it is also possible to reason deductively
about the function because each definition extends the underlying
logic with a definitional axiom. To ensure that the logic is sound
after the addition of this axiom, certain restrictions have to be
met, namely that the recursion terminates. This can be quite
challenging.

Because ACL2 is a programming language, you often may wish simply to
program in ACL2. For example, you may wish to define your system
and test it, without any logical burden. Or, you may wish to define
``utility'' functions -- functions that are executed to help manage
the task of building your system but functions whose logical
properties are of no immediate concern. Such functions might be
used to generate test data or help interpret the results of tests.
They might create files or explore the ACL2 data base. The
termination arguments for such functions are an unnecessary burden
provided no axioms about the functions are ever used in deductions.

We think of functions having :program mode as ``dangerous''
functions, while functions having :logic mode are ``safe.'' The
only requirement enforced on :program mode functions is the
syntactic one: each definition must be well-formed ACL2. Naively
speaking, if a :program mode function fails to terminate then no
harm is done because no axiom is added (so inconsistency is avoided)
and some invocations of the function may simply never return. This
simplistic justification of :program mode execution is faulty
because it ignores the damage that might be caused by
``mis-guarded'' functions. See defun-mode-caveat.

We therefore implicitly describe an imagined implementation of
defun-modes that is safe and, we think, effective. But please
see defun-mode-caveat.

The default defun-mode is :logic. This means that when you defun a
function the system will try to prove termination. If you wish to
introduce a function of a different defun-mode use the :modexargs
keyword. Below we show fact introduced as a function in :program
mode.

No axiom is added to the logic as a result of this definition. By
introducing fact in :program mode we avoid the burden of a
termination proof, while still having the option of executing the
function. For example, you can type

ACL2 !>(fact 3)

and get the answer 6. If you type (fact -1) you will get a hard
lisp error due to ``infinite recursion.''

However, the ACL2 theorem prover knows no axioms about fact. In
particular, if the term (fact 3) arises in a proof, the theorem
prover is unable to deduce that it is 6. From the perspective of
the theorem prover it is as though fact were an undefined
function symbol of arity 1. Thus, modulo certain important
issues (see defun-mode-caveat), the introduction of this
function in :program mode does not imperil the soundness of the
system -- despite the fact that the termination argument for fact
was omitted -- because nothing of interest can be proved about
fact. Indeed, we do not allow fact to be used in logical
contexts such as conjectures submitted for proof.

It is possible to convert a function from :program mode to
:logic mode at the cost of proving that it is admissible. This can
be done by invoking

(verify-termination fact)

which is equivalent to submitting the defun of fact, again, but
in :logic mode.

This particular event will fail because the termination argument requires
that n be nonnegative. A repaired defun, for example with =
replaced by <=, will succeed, and an axiom about fact will
henceforth be available.

Technically, verify-termination submits a redefinition of the
:program mode function. This is permitted, even when
ld-redefinition-action is nil, because the new definition is
identical to the old (except for its :mode and, possibly, other
non-logical properties).

See guard for a discussion of how to restrict the execution of
functions. Guards may be ``verified'' for functions in :logic
mode; see verify-guards.